Finite Group Databasebeta

← #28558374

Subgroups — #28558374

Order 15104.

678 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433179C211yesmaximal, characteristic, complemented
2#21433120C211yesmaximal, characteristic
3C2×C4×#11745670C211yesmaximal, characteristic
4#21422950C214yesmaximal, complemented
5#21422929C214yesmaximal, complemented
6#21422842C211yesmaximal, characteristic, complemented
7#21422492C211yesmaximal, characteristic, complemented
8#21422389C211yesmaximal, characteristic, complemented
9C59×#1478C211yesmaximal, characteristic
10C22×#11745670C411yescharacteristic
11C2×C4×Dic118C2211yescharacteristic
12#16837532—22no—
13#16837529C2214yescomplemented
14C4×#11745670C412yes—
15C4×#11745670C412yes—
16C4×#11745670C2212yes—
17C4×#11745670C2212yes—
18D4×C59⋊C8—22no—
19D4×C59⋊C8C2214yes—
20#16837456—22no—
21#16837455C2211yescharacteristic
22#16837455C2211yescharacteristic
23C2×#14273658C411yescharacteristic
24C2×#14273650C411yescharacteristic
25C2×#14273648C411yescharacteristic
26C2×C22⋊(C59⋊C8)—22no—
27C2×C4×C59⋊C8C2211yescharacteristic
28#16837154C2212yescomplemented
29#16837152C2212yescomplemented
30#16837134C2214yescomplemented
31#16837124C2212yescomplemented
32#16837123C2212yescomplemented
33C2×C4×D4×C59C2211yescharacteristic
34C59×#431C2211yescharacteristic
35C59×#431C2211yescharacteristic
36C59×D4⋊C8C2214yes—
37C22×Dic118C2×C411yescharacteristic
38C2×#11745670—24no—
39C2×#11745670C2×C412yes—
40C2×#11745670C2×C412yes—
41C22×C59⋊C8—24no—
42C22×C59⋊C8C2×C411yescharacteristic
43C22×C59⋊C8C2×C411yescharacteristic
44C4×Dic118—21no—
45C4×Dic118—21no—
46C4×Dic118C2×C412yes—
47C4×Dic118C2312yes—
48C2×C4⋊Dic59C2×C411yescharacteristic
49C2×Dic59⋊C4—21no—
50C2×C4×Dic59—21no—
51#14273658C2×C412yes—
52#14273658C2×C412yes—
53C4⋊(C59⋊C8)D412yes—
54C4⋊(C59⋊C8)C2312yes—
55Dic59⋊C8—24no—
56#14273650C2×C412yes—
57#14273650C2×C412yes—
58#14273648C2×C412yes—
59#14273648C2×C412yes—
60C22⋊(C59⋊C8)—21no—
61C22⋊(C59⋊C8)—24no—
62C22⋊(C59⋊C8)D412yes—
63#14273637—24no—
64C4×C59⋊C8—21no—
65C4×C59⋊C8C2×C411yescharacteristic
66C4×C59⋊C8C2×C411yescharacteristic
67C4×C59⋊C8C2311yescharacteristic
68C4×C59⋊C8C2311yescharacteristic
69C4×C59⋊C8C2312yes—
70C22×D4×C59C2×C411yescharacteristic
71C23×C4×C59—21no—
72C4×D4×C59—21no—
73C4×D4×C59—21no—
74C4×D4×C59C2×C412yes—
75C4×D4×C59C2312yes—
76C2×C4⋊C4×C59C2×C411yescharacteristic
77C2×C22⋊C4×C59—21no—
78C2×C42×C59C2311yescharacteristic
79C4⋊C8×C59—21no—
80C4⋊C8×C59C2312yes—
81C22⋊C8×C59—21no—
82C22⋊C8×C59—21no—
83C4×C8×C59—21no—
84C4×C8×C59C2312yes—
85C24×C59—21no—
86C2×D4×C59—21no—
87C2×D4×C59—22no—
88C2×D4×C59M1614yes—
89C2×D4×C59C22×C412yes—
90C22×C4×C59—21no—
91C22×C4×C59—22no—
92C22×C4×C59—22no—
93C22×C4×C59—82no—
94C22×C4×C59C22×C411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59#4111yescharacteristic
97C22×Dic59—21no—
98C22×Dic59—21no—
99C2×Dic118—22no—
100C2×Dic118—22no—
101C2×Dic118—21no—
102C2×Dic118C22×C412yes—
103C2×C8×C59—22no—
104C2×C8×C59—22no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59C22×C412yes—
107C22⋊C4×C59—22no—
108C22⋊C4×C59—22no—
109C42×C59C22×C411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59#4111yescharacteristic
112C42×C59C2411yescharacteristic
113#11745670—24no—
114#11745670—24no—
115C4⋊Dic59—21no—
116C4⋊Dic59C22×C412yes—
117Dic59⋊C4—22no—
118Dic59⋊C4—22no—
119C4×Dic59—22no—
120C4×Dic59—22no—
121C2×C59⋊C8—22no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—84no—
125C2×C59⋊C8C22×C411yescharacteristic
126C2×C59⋊C8C22×C411yescharacteristic
127C2×C59⋊C8C22×C411yescharacteristic
128C2×C59⋊C8C22×C411yescharacteristic
129C2×C59⋊C8C2×D414yes—
130C23×C59—22no—
131C23×C59—24no—
132C23×C59—81no—
133C23×C59—81no—
134C23×C59C4×D411yescharacteristic
135D4×C59—22no—
136D4×C59—24no—
137D4×C59C2×M1614yes—
138C2×C4×C59—22no—
139C2×C4×C59—22no—
140C2×C4×C59—41no—
141C2×C4×C59—44no—
142C2×C4×C59—82no—
143C2×C4×C59—81no—
144C2×C4×C59C4×D411yescharacteristic
145C2×C4×C59C4×D411yescharacteristic
146C2×C4×C59C4×D411yescharacteristic
147C2×C4×C59C2×M1612yes—
148C2×C4×C59#13112yes—
149C2×C4×C59C23×C411yescharacteristic
150C2×C4×C59C22×D411yescharacteristic
151C2×C4×C59C2×#4111yescharacteristic
152C2×Dic59—22no—
153C2×Dic59—22no—
154C2×Dic59—42no—
155C2×Dic59—81no—
156Dic118—22no—
157Dic118—22no—
158Dic118—42no—
159C8×C59—42no—
160C8×C59—42no—
161C59⋊C8—21no—
162C59⋊C8—21no—
163C59⋊C8—21no—
164C59⋊C8—21no—
165C59⋊C8—24no—
166C59⋊C8—42no—
167#17879—591nomaximal, Sylow
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—24no—
171C22×C59—42no—
172C22×C59—42no—
173C22×C59—84no—
174C22×C59#43512yes—
175C22×C59C2×C4×D411yescharacteristic
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—21no—
179C4×C59—42no—
180C4×C59#43512yes—
181C4×C59C22×M1611yescharacteristic
182C4×C59C2×#13111yescharacteristic
183C4×C59#57712yes—
184Dic59—42no—
185Dic59—42no—
186#2983—591no—
187#2972—591no—
188C2×C4×SD16—591no—
189#1585—594no—
190#1582—594no—
191#1564—591no—
192#1479—591no—
193#1478D5911yescharacteristic
194#1452—591no—
195C2×C59—22no—
196C2×C59—44no—
197C2×C59C2×#43511yescharacteristic
198C2×C59#294911yescharacteristic
199C2×C59#295711yescharacteristic
200C22×SD16—591no—
201C2×C4×Q8—591no—
202C2×C4×D4C2×D5911yescharacteristic
203#446—594no—
204#446—1182no—
205C4×SD16—592no—
206C4×SD16—592no—
207C4×SD16—592no—
208C4×SD16—592no—
209C8×D4—594no—
210C8×D4—1182no—
211#432—1182no—
212#431—591no—
213#431—591no—
214#431C2×D5911yescharacteristic
215#431C2×D5911yescharacteristic
216C2×#106—591no—
217C2×#103—591no—
218C2×#102—591no—
219C2×C22⋊C8—1182no—
220C2×C4×C8—591no—
221#335—592no—
222#334—592no—
223Q8⋊C8—594no—
224D4⋊C8C2×D5914yes—
225#322—592no—
226#322—592no—
227C59#1787911yescharacteristic, complemented, Sylow
228C22×Q8—591no—
229C22×D4C4×D5911yescharacteristic
230C23×C4—21no—
231C2×SD16—592no—
232C2×SD16—592no—
233C2×SD16—1184no—
234C22×C8—591no—
235C22×C8—591no—
236C22×C8—1184no—
237C4×Q8—592no—
238C4×Q8—592no—
239C4×Q8—1181no—
240C4×Q8—1181no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4C4×D5912yes—
244C4×D4C22×D5912yes—
245C2×C4⋊C4—591no—
246C2×C4⋊C4—1181no—
247C2×C4⋊C4C4×D5911yescharacteristic
248C2×C22⋊C4—21no—
249C2×C42—1181no—
250C2×C42C22×D5911yescharacteristic
251#106—592no—
252#106—592no—
253C4⋊C8—21no—
254C4⋊C8—592no—
255C4⋊C8—592no—
256C4⋊C8—1184no—
257C4⋊C8C22×D5912yes—
258#103—592no—
259#103—592no—
260#102—592no—
261#102—592no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—592no—
265C22⋊C8—1184no—
266C22⋊C8—1181no—
267#97—1184no—
268C4×C8—21no—
269C4×C8—591no—
270C4×C8—591no—
271C4×C8—591no—
272C4×C8—591no—
273C4×C8—592no—
274C4×C8—1181no—
275C4×C8C22×D5912yes—
276C24—21no—
277C2×Q8—592no—
278C2×Q8—1181no—
279C2×Q8—1182no—
280C2×Q8—1182no—
281C2×D4—21no—
282C2×D4—22no—
283C2×D4#1174565914yes—
284C2×D4C2×C4×D5912yes—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—82no—
289C22×C4—1181no—
290C22×C4—1181no—
291C22×C4C2×C4×D5911yescharacteristic
292C22×C4D4×D5911yescharacteristic
293C22×C4#1174568711yescharacteristic
294SD16—1184no—
295SD16—1184no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—591no—
299C2×C8—591no—
300C2×C8—591no—
301C2×C8—591no—
302C2×C8—594no—
303C2×C8—1182no—
304C2×C8—1182no—
305C2×C8—1182no—
306C2×C8—4724no—
307C4⋊C4—21no—
308C4⋊C4—592no—
309C4⋊C4—1181no—
310C4⋊C4—1182no—
311C4⋊C4—1182no—
312C4⋊C4C2×C4×D5912yes—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C42—1182no—
316C42—1182no—
317C42C2×C4×D5911yescharacteristic
318C42D4×D5911yescharacteristic
319C42#1174568711yescharacteristic
320C42C23×D5911yescharacteristic
321C23—22no—
322C23—24no—
323C23—81no—
324C23—81no—
325C23(C22⋊D59)⋊C411yescharacteristic
326Q8—1182no—
327Q8—1182no—
328Q8—2362no—
329D4—22no—
330D4—24no—
331D4C2×#1174565914yes—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—1182no—
339C2×C4—1182no—
340C2×C4—4721no—
341C2×C4—2362no—
342C2×C4(C22⋊D59)⋊C411yescharacteristic
343C2×C4(C22⋊D59)⋊C411yescharacteristic
344C2×C4(C22⋊D59)⋊C411yescharacteristic
345C2×C4C2×#1174565912yes—
346C2×C4#1427373512yes—
347C2×C4C22×C4×D5911yescharacteristic
348C2×C4C2×D4×D5911yescharacteristic
349C2×C4C2×#1174568711yescharacteristic
350C8—42no—
351C8—42no—
352C8—1181no—
353C8—1181no—
354C8—1181no—
355C8—1181no—
356C8—1184no—
357C8—2362no—
358C22—21no—
359C22—21no—
360C22—24no—
361C22—42no—
362C22—42no—
363C22—84no—
364C22#1683747012yes—
365C22C2×(C22⋊D59)⋊C411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—2362no—
371C4—2362no—
372C4#1683747012yes—
373C4C22×#1174565911yescharacteristic
374C4C2×#1427373511yescharacteristic
375C4#1683821212yes—
376C2—22no—
377C2—44no—
378C2C2×#1683747011yescharacteristic
379C2#2143295811yescharacteristic
380C2#2143300911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1478C2
socleC22×C59C2×C4×D4
prime radicalC22×D4×C59C2×C4
cosocleC42×C59C24
hypercenter#1478D59
hypocenterC59#17879
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1478

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21433179C2semidirect
4. #21422950C2semidirect
5. #21422929C2semidirect
6. #21422842C2semidirect
7. #21422492C2semidirect
8. #21422389C2semidirect
13. #16837529C22semidirect
28. #16837154C22semidirect
29. #16837152C22semidirect
30. #16837134C22semidirect
31. #16837124C22semidirect
32. #16837123C22semidirect
227. C59#17879semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214331792
#214331202
C2×C4×#117456702
#214229502
#214229292
#214228422
#214224922
#214223892
C59×#14782
#1787959